Why must we find a common denominator when add/subtract fractions?
If you have fractions with different denominators, thn it's kinda like comparing apples and oranges.
By finding a common denominator, you convert both fractions into something that can be compared.
e.g.
2/5 + 1/2
= 4/10 + 5/10
= 9/10
Without finding that common denominator, you wouldn't be able to add the numbers together.
If you don't, it is like trying to add or subtract apples and oranges... Or monkeys and kideny beans.
Suppose you cut a pizza into 8 pieces. Each piece is an "eighth" of a pizza. If you add two pieces to one piece, you have three-eigths of the pizza. You cannot add 1/8 and 2/8 and get 3/16. Only the numerators get added, and the denominators must be the same when you do this.
When adding or subtracting fractions, it is necessary to find a common denominator because fractions with different denominators cannot be directly added or subtracted. Finding a common denominator ensures that the fractions can be combined by having the same denominator.
To understand why this is necessary, let's start with an example. Suppose we want to add the fractions 1/3 and 1/4:
1/3 + 1/4
To add these fractions, we need them to have a common denominator. The common denominator is the least common multiple (LCM) of the original denominators, which in this case is 12.
To find a common denominator, we can follow these steps:
1. Identify the denominators of the fractions (in this case, 3 and 4).
2. Find the LCM of the denominators (in this case, 12).
3. Rewrite the fractions using the LCM as the new denominator, while maintaining their value. Multiply the numerator and denominator of each fraction by a factor that makes the denominator equal to the LCM.
- For 1/3: Multiply both the numerator and denominator by 4, so it becomes 4/12.
- For 1/4: Multiply both the numerator and denominator by 3, so it becomes 3/12.
Now, we have both fractions with a common denominator of 12:
4/12 + 3/12
Since the denominators are the same, we can simply add the numerators:
(4 + 3)/12 = 7/12
So, the sum of 1/3 and 1/4 is 7/12.
By finding a common denominator, we ensure that the fractions have the same base and can be added or subtracted accurately. This adjustment allows us to perform the arithmetic operation correctly and obtain the sum or difference of the fractions.